Whitehead groups of certain semidirect products of free groups

Author:
Koo Guan Choo

Journal:
Proc. Amer. Math. Soc. **43** (1974), 26-30

MSC:
Primary 18F25

DOI:
https://doi.org/10.1090/S0002-9939-1974-0338124-4

MathSciNet review:
0338124

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Abstract: Let be a direct product of free groups an automorphism of which leaves all but one of the noncyclic factors in pointwise fixed, an infinite cyclic group and another free group. Let be the semidirect product of and with respect to and the semidirect product of and with respect to the automorphism of induced by . We prove that the Whitehead group of and the projective class group of the integral group ring are trivial. These results extend that of [**3**].

**[1]**H. Bass, A. Heller and R. G. Swan,*The Whitehead group of a polynomial extension*, Inst. Hautes Études Sci. Publ. Math. No. 22 (1964), 61-79. MR**30**#4806. MR**0174605 (30:4806)****[2]**K. G. Choo,*Whitehead groups of twisted free associative algebras*, Pacific J. Math. (to appear). MR**0374175 (51:10375)****[3]**-*The projective class group of the fundamental group of a surface is trivial*, Proc. Amer. Math. Soc.**40**(1973), 42-46. MR**0323869 (48:2222)****[4]**F. T. Farrell and W. C. Hsiang,*A formula for*, Applications for Categorical Algebra, Proc. Sympos. Pure Math., vol., 17, Amer. Math. Soc., Providence, R.I., 1970, pp. 192-218. MR**41**#5457. MR**0260836 (41:5457)****[5]**J. Stallings,*Whitehead torsion of free products*, Ann. of Math. (2)**82**(1965), 354-363. MR**31**#3518. MR**0179270 (31:3518)**

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0338124-4

Keywords:
Whitehead group,
projective class group,
semidirect product of free groups,
twisted group ring

Article copyright:
© Copyright 1974
American Mathematical Society